So guys I need some help with these types of exercices. I am not sure if my proof is right and if so can I use it also for the others?
$Let \ A\subset \mathbb{R} \ and \ B\subset \mathbb{R} \ we \ define \ C= \left \{ a+b: a \epsilon A,b\epsilon B \right \} \ We \ want \ to \ prove:$
1) Inf C = Inf A + Inf B
2) sup C = sup A + sup B
3) Max C = Max A + Max B
4) Min C = Min A + Min B
Proof of 2) : Let us first show that A+B is bounded above given that A and B are bounded above. Since A and B are bounded above, there exist α ∈ R and β ∈ R such that a ≤ α for all a ∈ A, b ≤ β for all b ∈ B. For any x ∈ A+B, there exist a ∈ A and b ∈ B such that x = a+b.
Thus, x = a+b ≤ α +β, which shows that A+B is bounded above. Moreover, since supA is an upper bound of A and supB is an upper bound of B, we have
x = a+b ≤ supA+supB. This implies that supA+supB is an upper bound of A+B and, hence,
sup(A+B) ≤ supA+supB.
Fix any ε > 0, there exits a ∈ A and b ∈ B such that
supA−ε < a and
supB−ε < b.
It follows that supA+supB−2ε < a+b ≤ sup(A+B). Since ε > 0 is arbitrary,
supA+supB ≤ sup(A+B). Therefore, the given equality has been justified.
